Definition 96.24.1. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. The associated affine site is the full subcategory $\mathcal{X}_{affine}$ of $\mathcal{X}$ whose objects are those $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over a scheme $U$ such that $U$ is affine. The topology on $\mathcal{X}_{affine}$ will be the chaotic one, i.e., such that sheaves on $\mathcal{X}_{affine}$ are the same as presheaves.
96.24 Restricting to affines
In this section, given a category $\mathcal{X}$ fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ we will consider the full subcategory $\mathcal{X}_{affine}$ of $\mathcal{X}$ consisting of objects $x$ lying over affine schemes $U$. We will see how, for any topology $\tau $ finer than the Zariski topology, the category of sheaves on $\mathcal{X}$ and $\mathcal{X}_{affine, \tau }$ agree.
Thus the functor $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ restricts to a functor
\[ p : \mathcal{X}_{affine} \longrightarrow (\textit{Aff}/S)_{fppf} \]
where the notation on the right hand side is the one introduced in Topologies, Definition 34.7.8. It is clear that $\mathcal{X}_{affine}$ is fibred in groupoids over $(\textit{Aff}/S)_{fppf}$. It follows that $\mathcal{X}_{affine}$ inherits a Zariski, étale, smooth, syntomic, and fppf topology from $(\textit{Aff}/S)_{Zar}$, $(\textit{Aff}/S)_{\acute{e}tale}$, $(\textit{Aff}/S)_{smooth}$, $(\textit{Aff}/S)_{syntomic}$, and $(\textit{Aff}/S)_{fppf}$, see Stacks, Definition 8.10.2.
Definition 96.24.2. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids.
The associated affine Zariski site $\mathcal{X}_{affine, Zar}$ is the structure of site on $\mathcal{X}_{affine}$ inherited from $(\textit{Aff}/S)_{Zar}$.
The associated affine étale site $\mathcal{X}_{affine, {\acute{e}tale}}$ is the structure of site on $\mathcal{X}_{affine}$ inherited from $(\textit{Aff}/S)_{\acute{e}tale}$.
The associated affine smooth site $\mathcal{X}_{affine, smooth}$ is the structure of site on $\mathcal{X}_{affine}$ inherited from $(\textit{Aff}/S)_{smooth}$.
The associated affine syntomic site $\mathcal{X}_{affine, syntomic}$ is the structure of site on $\mathcal{X}_{affine}$ inherited from $(\textit{Aff}/S)_{syntomic}$.
The associated affine fppf site $\mathcal{X}_{affine, fppf}$ is the structure of site on $\mathcal{X}_{affine}$ inherited from $(\textit{Aff}/S)_{fppf}$.
This definition makes sense by the discussion above. For each $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $ a family of morphisms $\{ x_ i \to x\} _{i \in I}$ with fixed target in $\mathcal{X}_{affine}$ is a covering in $\mathcal{X}_{affine, \tau }$ if and only if the family of morphisms $\{ p(x_ i) \to p(x)\} _{i \in I}$ of affine schemes is a standard $\tau $-covering as defined in Topologies, Definitions 34.3.4, 34.4.5, 34.5.5, 34.6.5, and 34.7.5.
Lemma 96.24.3. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. The functor $\mathcal{X}_{affine, \tau } \to \mathcal{X}_\tau $ is a special cocontinuous functor. Hence it induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{affine, \tau })$ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau )$.
Proof. Omitted. Hint: the proof is exactly the same as the proof of Topologies, Lemmas 34.3.10, 34.4.11, 34.5.9, 34.6.9, and 34.7.11. $\square$
Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let us denote $\mathcal{O}$ the restriction of $\mathcal{O}_\mathcal {X}$ to $\mathcal{X}_{affine}$. Then $\mathcal{O}$ is a sheaf in the Zariski, étale, smooth, syntomic, and fppf topologies on $\mathcal{X}_{affine}$. Furthermore, the equivalence of topoi of Lemma 96.24.3 extends to an equivalence
96.24.3.1
\begin{equation} \label{stacks-sheaves-equation-alternative-ringed} (\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{affine, \tau }), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau ), \mathcal{O}_\mathcal {X}) \end{equation}
of ringed topoi for $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $.
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